On the number of invariant measures for random expanding maps in higher dimensions

Abstract

In [22], Jab loński proved that a piecewise expanding C 2 multidimensional Jab loński map admits an absolutely continuous invariant probability measure (ACIP). In [6], Boyarsky and Lou extended this result to the case of i.i.d. compositions of the above maps, with an on average expanding condition. We generalize these results to the (quenched) setting of random Jab loński maps, where the randomness is governed by an ergodic, invertible and measure preserving transformation. We prove that the skew product associated to this random dynamical system admits a finite number of ergodic ACIPs. Furthermore, we provide two different upper bounds on the number of mutually singular ergodic ACIPs, motivated by the works of Buzzi [9] in one dimension and Góra, Boyarsky and Proppe [19] in higher dimensions.

Invariant measures for random expanding on average Saussol maps

“…This work can be seen as a generalization of the work in [3] on random Jab loński maps where each component of the map only depends on its corresponding variable. In this paper, we include maps such that the components are allowed to depend on all or some of the variables.…”

“…In this paper, we include maps such that the components are allowed to depend on all or some of the variables. In [3], the authors studied the quenched setting of random Jab loński maps. They proved that the skew product associated to this random dynamical system admits a finite number of ergodic ACIPs.…”

“…In Remark 2.3 in [19], Liverani gave a sufficient condition for the uniqueness of ACIP using dense orbits. However, in spite of the fact that Batayneh and González-Tokman [3], Buzzi [5] and Araujo-Solano [2] provided bounds on the number of ACIPs for random compositions of certain classes of one and multidimensional maps, studying and, particularly, bounding the number of ergodic ACIPs is still largely open problem. This relates to questions about multiplicity of Lyapunov exponents in multiplicative ergodic theory.…”